1. 25-26 January Bochum, Germany Luc Hantcherli - Philip Eisenlohr - Franz Roters – Dierk Raabe and Collaboration between: Mechanical twinning in crystal plasticity finite element methods 7th GAMM Seminar on Microstructures

2. Slide 2 Introduction undeformed 10 % strain 50 % strain 30 % strain

3. Slide 3 Outline of the presentation Part 1: Basics of Crystal Plasticity Finite Element Modeling (CPFEM) Part 2: Review of Kalidindi’s phenomenological approach to mechanical twinning Part 3: Introduction to a more physically-based approach to mechanical twinning Part 4: Results and discussion Part 5: Conclusions and outlook

4. Slide 4 1-Basics of Crystal Plasticity Finite Element Modeling Continuum mechnics / FEM: Space and time discretization Notion of integration point Reference Configuration Current Configuration Intermediate stress-free Configuration FeFe FpFp F Continuum mechanics: Notion of tensors Multiplicative decomposition F = F e F p

5. Slide 5 1-Basics of Crystal Plasticity Finite Element Modeling Constitutive equations: Hooke law - T* = C:E* Reference Configuration Current Configuration Intermediate stress-free Configuration FeFe FpFp F Flow rule: (given here for F p ) F p = L p F p

6. Slide 6 1-Basics of Crystal Plasticity Finite Element Modeling Current Configuration Reference Configuration Intermediate stress-free Configuration F FpFp FeFe Homogeneization: Taylor-type Hooke law T* = C hom :E* Crystal Plasticity: Notion of kinematics, i.e. finite number of possible deformation modes Description of L p

7. Slide 7 1-Basics of Crystal Plasticity Finite Element Modeling Slip deformation in the parent region Twin formation from the parent region Model for slip: Flow rule and Hardening rule give Model for twin: “Flow rule” and “Hardening rule” give

8. Slide 8 Outline of the presentation Part 1: Basics of Crystal Plasticity Finite Element Modeling (CPFEM) Part 2: Review of Kalidindi’s phenomenological approach to mechanical twinning Part 3: Introduction to a more physically-based approach to mechanical twinning Part 4: Discussions on the proposed models Part 5: Conclusions and outlook

9. Slide 9 2-Review of Kalidindi’s phenomenological approach to mechanical twinning Model initially proposed by S. Kalidindi (Kalidindi 2001) Flow rule for slip: - 12 reduced slip systems - a viscoplastic power-type law - a CRSS-based activation Flow rule for twin: - 12 twin systems - a power-type law - a unidirectional CRSS-based activation assumed analogy

10. Slide 10 2-Review of Kalidindi’s phenomenological approach to mechanical twinning Twins do not contribute to an extra-hardening for coplanar slip systems Twins contribute to an extra- hardening for non-coplanar slip systems

11. Slide 11 2-Review of Kalidindi’s phenomenological approach to mechanical twinning

12. Slide 12 2-Review of Kalidindi’s phenomenological approach to mechanical twinning Geometry/Mesh - 125 linear cubic elements, each with 8 integration points - periodic boundary conditions - 10 random orientations per integration point (Taylor homogenization) -deformation in unidirectional tension

13. Slide 13 2-Review of Kalidindi’s phenomenological approach to mechanical twinning 8515035526570010 4 8000 [MPa]

14. Slide 14 Outline of the presentation Part 1: Basics of Crystal Plasticity Finite Element Modeling (CPFEM) Part 2: Review of Kalidindi’s phenomenological approach to mechanical twinning Part 3: Introduction to a more physically-based approach to mechanical twinning Part 4: Results ans discussion Part 5: Conclusions and outlook

15. Slide 15 3-Introduction to a more physically-based approach to mechanical twinning Some ideas initially proposed by S. Allain (Phd thesis 2004) introduce more physically-based variables, e.g. dislocation densities 1 st idea: 2 nd idea: consider deformation twinning as nucleation- growth process 3 rd idea: deeper explore the morphological and topological properties of microstructure

16. Slide 16 3-Introduction to a more physically-based approach to mechanical twinning Physically-based state variables: Introduction of, immobile dislocation dentisity per glide system Derivation of 3 populations of dislocations:, and Flow rule: Description of the shear rates using mobile dislocation densities and corresponding velocities (Orowan equation) Hardening rule: Evolution of the immobile dislocation densities from multiplication and recovery rates

17. Slide 17 3-Introduction to a more physically-based approach to mechanical twinning Requirements for the twin nucleation law: Need of special dislocation configurations, e.g. locks, as preferential sites for twin nucleation volume density of dislocation reactions Need of local stress increase on these configurations, e.g. pile-ups, to trigger the formation of a twin nucleus volume fraction sampled for building pile-ups Need of a Schmid criterion based nucleation, e.g. classical power-law Final expression for twin nucleation law: Volume density of activated twin nuclei through expressed as:

18. Slide 18 3-Introduction to a more physically-based approach to mechanical twinning dA d*d* slip plane dislocation lines dislocation reactions capture volume

19. Slide 19 3-Introduction to a more physically-based approach to mechanical twinning system β system β‘ dβ‘dβ‘ mfp β eβeβ Twin volume fraction evolution: Computation assuming a recrystallisation like behaviour and instantaneous growth of the freshly nucleated twins: with

20. Slide 20 Outline of the presentation Part 1: Basics of Crystal Plasticity Finite Element Modeling (CPFEM) Part 2: Review of Kalidindi’s phenomenological approach to mechanical twinning Part 3: Introduction to a more physically-based approach to mechanical twinning Part 4: Results and discussion Part 5: Conclusions and outlook

21. Slide 21 4-Results and discussion

22. Slide 22 4-Results and discussion Increase of d*

23. Slide 23 4-Results and discussion decrease of C4

24. Slide 24 4-Results and discussion Advantages: - Introduction of relevant variables, e.g. grain size, temperature, stacking fault energies - Dislocation-based twin nucleation law Drawbacks: - Lost of computational efficiency, long calculation time, numerical instabilities - Crystal plasticity induced limitation, e.g. use of continuous and derivable equations

25. Slide 25 Outline of the presentation Part 1: Basics of Crystal Plasticity Finite Element Modeling (CPFEM) Part 2: Review of Kalidindi’s phenomenological approach to mechanical twinning Part 3: Introduction to a more physically-based approach to mechanical twinning Part 4: Results and discussion Part 5: Conclusions and outlook

26. Slide 26 5-Conclusions and outlook Conclusion: We proposed a physically-based CPFEM modeling that capture some of the main physics of mechanical twinning shown in TWIP steels. Advantages and drawbacks were discussed. 2 ways for future works: –To try to pursue the modeling of mechanical twinning, including some new features like no constant twin thickness, new deformation modes that allow twins to deform plastically –To start microstructural investigations of TWIP steels, with particular focus on the nucleation of mechanical twins

27. Slide 27 5-Conclusions and outlook Thank you all for your attention!